Jawaharlal Nehru Technological University Kakinada 2007 B.Tech Computer Science and Engineering PROBABILITY THEORY AND STOCHASTIC PROCESS (3) - Question Paper

Code No: R059210401 Set No. 2
II B.Tech I Semester Regular Examinations, November 2007
PROBABILITY THEORY AND STOCHASTIC PROCESS
( Common to Electronics & Communication Engineering, Electronics &
Telematics and Electronics & Computer Engineering)
Time: three hours Max Marks: 80
ans any 5 ques.
All ques. carry equal marks
? ? ? ? ?
1. (a) With an example describe and discuss the following:
i. Equality likely events
ii. Exhaustive events.
iii. Mutually exclusive events.
(b) In an experiment of picking up a resistor with identical likelihood of being picked
up for the events; A as “draw a 47 resistor”, B as “draw a resistor with
5% tolerance” and C as “draw a 100 resistor” from a box containing 100
resistors having resistance and tolerance as shown beneath. Determine joint
probabilities and conditional probabilities. [6+10]
Table 1
Number of resistor in a box having provided resistance and tolerance.
Resistance() Tolerance
5% 10% Total
22 10 14 24
47 28 16 44
100 24 eight 32
Total 62 38 100
2. (a) What is binomial density function? obtain the formula for binomial distrbution
function.
(b) What do you mean by continuous and discrete random variable? explain the
condition for a function to be a random variable. [6+10]
3. (a) describe moment generating function.
(b) State properties of moment generating function.
(c) obtain the moment generating function about origin of the Poisson distribution.
[3+4+9]
4. (a) describe conditional distribution and density function of 2 random variables
X and Y
(b) The joint probability density function of 2 random variables X and Y is
provided by
f(x, y) = _ a(2x + y2) 0 _ x _ two , two _ y _ 4
0 elsewhere
. obtain
1 of 2
i. value of ‘a’
ii. P(X _ 1,Y > 3). [8+8]
5. (a) let Xi, i = 1,2,3,4 be 4 zero mean Gaussian random variables. Use the joint
characteristic function to show that E {X1 X2 X3 X4} = E[X1 X2] E[X3 X4]
+ E[X1X3]E[X2X4] + E[X2X3] E[X1X4]
(b) Show that 2 random variables X1 and X2 with joint pdf.
fX1X2(X1,X2) = 1/16 |X1|< 4, two < X2< four are independent and orthogonal.[8+8]
6. A random process Y(t) = X(t)- X(t +t ) is described in terms of a process X(t) that
is at lowest wide sense stationary.
(a) Show that mean value of Y(t) is 0 even if X(t) has a non Zero mean value.
(b) Show that sY2= 2[RXX(0) - RXX(t )]
(c) If Y(t) = X(t) +X(t + t ) obtain E[Y(t)] and sY 2. [5+5+6]
7. (a) If the PSD of X(t) is Sxx(? ). obtain the PSD of dx(t)
dt
(b) Prove that Sxx (? ) = Sxx (-? )
(c) If R(t ) = ae|by|. obtain the spectral density function, where a and b are con-
stants. [5+5+6]
8. (a) A Signal x(t) = u(t) exp (-at ) is applied to a network having an impulse
response h(t)= ? u(t) exp (-? t). Here a & ? are real positive constants.
obtain the network response? (6M)
(b) 2 systems have transfer functions H1( ?) & H2( ?). Show the transfer
function H(?) of the cascade of the 2 is H(? ) =H1( ?) H2 (? ).
(c) For cascade of N systems with transfer functions Hn(?) , n=1,2,.... .N show
that H( ?) = pHn(?). [6+6+4]

Earning:   Approval pending.